Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
9138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(1,2,3,4)(8,12)(10,11)(13,14)(15,18)(16,17)(19,20);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12);(6,7)(10,11)(14,17)(18,20);(13,14)(15,18)(16,17)(19,20);(5,7,9,6)(8,11,12,10);(5,8)(6,10)(7,11)(9,12)(13,15,16,19)(14,18,17,20)Polycyclic presentation:
2^9|a^2=g,e^2=i,f^2=h,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72233744, #72233763, #72234453, … |
| maximal normal subgroup of | #72233744, #72233763, #72234453, … |
| maximal quotient of | #72256763, #72261769, #72262100, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7584478);